Chinese
Journal of Aeronautics Chinese Journal of Aeronautics 23(2010) 430-437
www.elsevier.com/locate/cja
Parametric Optimization Design of Aircraft Based on Hybrid Parallel Multi-objective Tabu Search Algorithm Qiu Zhiping*, Zhang Yuxing School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China Received 22 October 2009; accepted 1 February 2010
Abstract For dealing with the multi-objective optimization problems of parametric design for aircraft, a novel hybrid parallel multi-objective tabu search (HPMOTS) algorithm is used. First, a new multi-objective tabu search (MOTS) algorithm is proposed. Comparing with the traditional MOTS algorithm, this proposed algorithm adds some new methods such as the combination of MOTS algorithm and “Pareto solution”, the strategy of “searching from many directions” and the reservation of good solutions. Second, this article also proposes the improved parallel multi-objective tabu search (PMOTS) algorithm. Finally, a new hybrid algorithm — HPMOTS algorithm which combines the PMOTS algorithm with the non-dominated sorting-based multi-objective genetic algorithm (NSGA) is presented. The computing results of these algorithms are compared with each other and it is shown that the optimal result can be obtained by the HPMOTS algorithm and the computing result of the PMOTS algorithm is better than that of MOTS algorithm. Keywords: aircraft design; conceptual design; multi-objective optimization; tabu search; genetic algorithm; Pareto optimal
1. Introduction1 Aircraft design is an iterative process of analyzing, integrating and decision making which is comprehensively utilizing the fruits of contemporary science and technology, systems engineering methodology and the engineering language to guide the manufacturing, testing and operating. Aircraft conceptual design belongs to the primary phase of aircraft development which lays the foundation for conducting detail design. The methods used in aircraft conceptual design include statistical method and systems design[1]. Aircraft conceptual design is performed on the basis of accurate and advanced engineering numerical methods as well as stable and efficient optimization algorithms, so the study and application of optimization algorithms become the key for the parametric design of aircraft. Because aircraft conceptual design is always the issue of multi-objective optimization, all the objectives will not be able to reach optimums at the same time, so we must make tradeoff between these objectives[2]. *Corresponding author. Tel.: +86-10-82323871. E-mail address:
[email protected] Foundation items: National Science Fund for Distinguished Young Scholars (10425208); Programme of Introducing Talents of Discipline to Universities (B07009) 1000-9361/$ - see front matter © 2010 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(09)60238-7
Traditional multi-objective optimization algorithms often combine several objectives into a single objective by a set of weights, and then the solutions can be obtained by single-objective optimization. Due to the absence of the necessary priori knowledge to choose suitable weights and being sensitive to the shape of “Pareto front”, the effectiveness of traditional multi-objective optimization algorithms is dissatisfactory. Until French economist V. Pareto[3] proposed the concept of “Pareto optimal”, the optimal solution of multi-objective optimization was extended from a point to a set. The concept of “Pareto solution set” is a milestone for the study of multi-objective optimization. Since existing algorithms cannot achieve the perfect “Pareto solution set”, we need to create some new method to solve multi-objective optimization effectively and obtain satisfactory noninferior set. 2. Multi-objective Tabu Search (MOTS) Algorithm Tabu search algorithm is a heuristic algorithm with strong ability of local search[4]. In order to avoid falling into the local optima, the algorithm has two significant features: move operation and tabu list. The new candidate solution is generated by moving operations in the neighborhood of the current solution. To avoid cycling, the recently visited moves are classified as forbidden and stored in a tabu list so that every step of the algo-
No.4
Qiu Zhiping et al. / Chinese Journal of Aeronautics 23(2010) 430-437
rithm is to choose the best move which is non-taboo. However, in some cases, if a tabu move improves upon the best solution found so far, then that move can be accepted. This is known as the aspiration criterion. Every complete cycle of tabu search algorithm is to determine the best candidate through selecting neighboring ones of the current solution, and then update the tabu list and designate the new candidate solution as the new current solution. The searching process continues like this until a predetermined stopping criterion is satisfied. In fact, tabu search algorithm is used to solve the single-objective combinatorial optimization problem[5] and then is extended into the areas of multi-objective combinatorial optimization and multi-objective optimization within the continuous space. In the multi-objective optimization, the most important task is to find the Pareto optimal solution set. In addition, to determine the optimal neighbor is also critical. In the neighborhood of each current solution, the fitness function of the identified optimal solution is often defined as the weighted average of objectives. As the weight of every objective is being updated dynamically, the Pareto front of multi-objective optimization problems may be found out. This article first provides a new tabu search algorithm, and then improves it and introduces another newer tabu search algorithm—hybrid tabu search algorithm combining the “Pareto front” -based multi-objective genetic algorithm with MOTS algorithm. The conclusions are then drawn through comparing the results of those algorithms. 2.1. A new MOTS algorithm
constraint is ⎧⎪0 d mj = ⎨ ⎪⎩ g j ( xm )
g j ( xm ) ≤ 0 Otherwise
Based on the fuzzy logic theory, the penalty values corresponding to different deviation ranges are 0 ≤ max(d m1 , d m 2 ,
⎧0 ⎪ ⎪1 ⎪ ⎪2 Rm = ⎨ ⎪3 ⎪ ⎪4 ⎪ ⎩5
Step 3 tion.
, d mn ) ≤ 0.01
0.01 < max(d m1 , d m 2 ,
, d mn ) ≤ 0.05
0.05 < max(d m1 , d m 2 ,
, d mn ) ≤ 0.10
0.10 < max(d m1 , d m 2 ,
, d mn ) ≤ 0.20
0.20 < max(d m1 , d m 2 ,
, d mn ) ≤ 0.60
0.60 < max(d m1 , d m 2 ,
, d mn ) ≤ 1.00
Determining the transferred fitness funcz k = f k′ + Rm
The determination of fitness functions lies on the weight vectors of each of the objectives. Two measures are provided to determine weight vector[7-8]: a) Each weight vector is set at λ =(λk, k =1,2, , K) where λk∈{0,1/r,…,(r−1)/r,1} (r is determined by the K
decision makers) and λk ≥ 0, and ∑ λk = 1 should be k =1
satisfied. b) For each objective k, nonnegative random weights λK ] is normalλk (λk ≥ 0) is set, then λ = [λ1 λ2 K
(1) Crucial steps and setting of parameters 1) Definition of fitness function In this article, the fitness function of multi-objective optimization is prescribed as K
fitness = ∑ λk zk k =1
where zk is the transformed individual objective function value of each X = [ x1 x2 xm ] , λk the corresponding weight of each objective, and K the number of objectives. To determine each objective zk (adding the effect of restrictions)，a fuzzy penalty function method[6] has been applied to transform the constrained multi-objective optimization problem into non-constrained one. The specific process is as follows: Step 1 Normalizing the objective functions.
f k′ =
· 431 ·
1 1 + fk
where fk is the individual objective function. Step 2 Defining the penalty value which is used to evaluating the extent of constraint violation. The resulting deviation of a point xm from the jth
ized and put ∑ λk = 1 . k =1
2) Generation of initial solution The procedure of generating the initial solution of tabu search algorithm is as follows: Step 1 Randomly generating an initial point within the specified range of each variable. Step 2 If Rm = 0 , turning to Step 3, otherwise turning to Step 1. Step 3 Setting the current point as a initial feasible solution. 3) Selection of neighborhood solution When the current value is set at x0=( xi0 ,i=1,2, , M), the neighborhood solutions will be selected based on the following principle[9]: xi = xi0 + R ( ximax − ximin ) / 10 n
where R is a random number within the interval [−1, 1], xi0 is the current value of variable xi, and the step length of variable is determined by the value of n which is depending on the calculation precision. In the process of exploring the neighborhood of the current point, some neighborhood solutions may be located outside the defined interval of independent variable, thus it is necessary to judge whether they are within the
· 432 ·
Qiu Zhiping et al. / Chinese Journal of Aeronautics 23(2010) 430-437
variable interval of the independent variable. If not, the solution will be deleted and a new neighborhood solution should be generated. 4) Combination of MOTS algorithm and “Pareto solution” Comparing with traditional optimization algorithm, the tabu search algorithm applied to multi-objective optimization must be bound to involve the mutual coordination among these objectives. In order to solve this problem, we must obtain the non-dominated sotulion set of the problem. The concept of “Pareto solution” is introduced into tabu searching so as to strive for finding the “Pareto front” of the problem which is better than the best solution for one certain objective. The implementation process is as follows: a) In every searching step for one certain weight vector, the neighborhood set of the current solution is always merged with the set of efficient solutions obtained in the former step, then the non-dominated solutions are sought from the union. b) To obtain the general non-dominated solution set for one certain weight vector, the non-dominated solution sets of all searching steps are always merged together, and then the non-dominated solution sets are sought from the union. c) To obtain the general non-dominated solution set for all weight vectors, the non-dominated solution sets corresponding to every weight vector are always merged together, then the non-dominated solutions are sought from the union. For a certain weight vector, the whole process of MOTS algorithm is shown as Fig.1. If we are searching
from different “directions” (namely choosing different weight vectors) simultaneously, then we must run through this process separately for different weight vectors. Finally, we can combine all the results and obtain the non-dominated solution set of the sum aggregate. 5) The strategy of “searching from many directions” In order to expand the searching area of tabu search and enhance the diversity of solutions, a new strategy of “searching from many directions” is adopted. We can set many “searching directions”, namely many weight vectors. A normal tabu searching is carried out for every weight vector, and then we merge the non-dominated solution sets corresponding with all weight vectors as follows: E = E0 ∪ E1 ∪ ∪ Es , where E is the set of efficient solutions. Obviously, comparing with one single weight vector, the sum aggregate must include more non-dominated solution sets. The flowchart is shown as Fig.2.
Fig.2
Fig.1
Module of MOTS algorithm.
No.4
Process of “searching from different directions”.
6) Reserving mechanism of good solutions a) In every searching step for one certain weight vector λv, the neighborhood set of the current solution is always merged with the optimal solution set to insure that the best solutions in the neighborhood set is being reserved. b) For one certain weight vector λv, the optimal solution sets of all searching steps are always merged together to insure the best solutions of the whole iterative process are not to be lost. c) For all weight vectors λv (v = 0,1,…,s), all the optimal solution sets corresponding to all weight vectors are always merged together to insure that the best solutions of the whole process are being included. (2) Running steps of algorithm (two objectives) Step 1 Initialization. Randomly generating an initial feasible solution πi; defining the efficient solutions set E=Φ and the tabu list TL=Φ.
No.4
Qiu Zhiping et al. / Chinese Journal of Aeronautics 23(2010) 430-437
· 433 ·
Step 2 Iterating. 1) π 0=πi. 2) Ev = {π 0 }, λv = v/s (where v = 0,1, , s , and s is a non-negative integer which is set in advance). 3) Cycling: a) randomly selecting N neighborhood solutions around π 0 (being not tabu moves and/or satisfying the aspiration criterion and establishing the neighborhood solution set N(π 0)={π u: u=1,2, ,N};
b) setting Ev = Ev ∪ N (π 0 ) , and removing all dominated solutions from Ev; c) for each π u, separately calculating the value of individual objective z1(π u) and z2(π u); d) selecting π u meeting the following conditions: *
zλv (π u ) = min *
u =1,2, , N
[λv z1 + (1 − λv ) z2 ]
(1) Fig.3
e) setting π =π , and updating TL; f) If Ncount
